2020 · Q4
Which one of the following is a set of solution of the equation x^(√x) = ⁿ√(x^x) if n is a positive integer?
Elementary MathematicsAlgebra
- A{1, n²}
- B{1, √n}
- C{1, n}
- D{n, n²}
Explanation
Take logarithms of both sides: √x·ln x = (x/n)·ln x. If ln x = 0, then x = 1 is a solution. Otherwise √x = x/n, i.e. n = √x, giving x = n²; indeed (n²)ⁿ = n²ⁿ = ⁿ√((n²)ⁿ²). Both values satisfy the equation for every positive integer n, so the solution set is {1, n²}.